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I have an expression that I would like to take limits of but not in a conventional sense. Take for example, an expression like

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This expression involves intermediate variables ${\cal X(\epsilon)},{\cal Y(\epsilon)}$ whose limits are dependent on how they behave as $\epsilon\to 0$. I would like to take limits of the above in such a way that ${\cal X},{\cal Y}\to 0$ but also that ${\cal X}/\epsilon,{\cal Y}/\epsilon\to\infty$ as $\epsilon\to 0$. This will allow the exponential term to decay to zero, whilst any term involving only ${\cal X}$ or ${\cal Y}$ will go to zero. Strictly speaking, the first term inside the second pair of brackets, ${\cal Y}/\epsilon$, diverges as $\epsilon \to 0$ but upon the action of multiplying by the prefactor $\epsilon$, it will decay to zero as $\epsilon\to0$.

The result I would like to obtain is $-4\epsilon\csc^{2}\theta$, which can be evaluated only on the knowledge of ${\cal X(\epsilon)},{\cal Y(\epsilon)}$ as described above. However, I don't actually want to take limits as $\epsilon \to 0$ of the whole expression. I want the result to still involve terms in $\epsilon$ given that the above is a term in an asymptotic expansion in $\epsilon$. If I take $\epsilon\to0$ along with $\mathcal{X}\to 0$ and $\mathcal{Y}\to 0$, I will just get zero, which is no use to me.

I am stumped on how to implement this in Mathematica, even though I know exactly what calculation I am doing. Are there any elegant ways I can accomplish this?

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    $\begingroup$ Maybe Series, with X[e] and Y[e] written as functions of epsilon? $\endgroup$
    – Michael E2
    Oct 14, 2020 at 18:21
  • $\begingroup$ Assuming X[e]~X0 e^alfa,Y[e]~Y0 e^beta as e->0,0<alfa,beta<1 (see MichaelE2's comment) gives a limit ~ 4 e^\[Beta] Y0 Csc[\[Theta]] $\endgroup$ Oct 15, 2020 at 6:35

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